Diffusion of calcium in biological tissues and accompanied mechano-chemical effects

Diffusion of calcium in biological tissues and accompanied mechano-chemical

effects

Zbigniew Peradzyoski & Bogdan Kaźmierczak

Institute of Applied Mathematics and Mechanics, University of Warsaw Banacha 2. 0-097 Warsaw. E-mail zperadz@mimuw.edu.pl and

Institute of Fundamental Technological Research, Polish Academy of Sciences Świętokrzyska 21, 0- 049 Warsaw

Diffusion of calcium and accompanying mechanical effects in three basic structures:

 In the bulk tissue – mathematically in a 3-dimentional space,

 In a thin tissue layer (basically 2-dimensional space)



In an infinite thin cylindrical volume (basically 1 –

dimensional space).

The diffusion of calcium plays an important role in the living cells. It is mainly manifested in the existence of waves of calcium concentration:

1. intracellular waves

2. across the tissue -extracellular waves.

 They are responsible for the coordination of the

response to the local changes of the conditions.

 Variation of calcium concentration can serve as a

mechanism of movement of cells (e.g. crawling

motility of keratocytes)

In supporting calcium waves the nonlinear, autocatalytic mechanism, represented by a bi-stable source term in the respective diffusion equation

𝑐 _𝑡 = 𝐷𝛥𝑐 + 𝑓(𝑐)

plays an important role.

Example: f = -𝐴 𝑐 − 𝑐 ₁ 𝑐 − 𝑐 ₂ 𝑐 − 𝑐 ₃ where 0 ≤ 𝑐 ₁ < 𝑐 ₂ < 𝑐 ₃ ,

The calcium wave represents a travelling front joining the

two stable equilibriums.

Experiments: calcium waves can be generated by local mechanical stimulation. So, there must be some coupling between the chemical (change of calcium concentration) and mechanical processes (deformation). Since the

deformations induced by traction are rather small, we assume that the tissue can be treated as a visco-elastic material, whose stress tensor has the following form:

𝜎 _𝑖𝑗 = 𝜆𝜃𝛿 _𝑖𝑗 + 2𝜇𝜖 _𝑖𝑗 + 𝜈 ₁ 𝜃 𝛿 _𝑖𝑗 + 𝜈 ₂ 𝜖 _𝑖𝑗 + 𝜏 _𝑖𝑗 , where λ and μ are the Lame coefficients, 𝜖 = (𝜖 _𝑖𝑗 ) is

the infinitesimal strain tensor, 𝜃 = 𝑇𝑟𝜖 is the dilation, ν 1 and ν 2 are the viscosities and 𝜏 = 𝜏 (𝑐) is the

symmetric traction tensor representing traction forces, generated by the sol- gel transition of the cytoplasm.

This transition is caused by the change of calcium concentration.

The diffusion of calcium is rather slow ( e.g. the speed of calcium waves is of the order of tens microns per second), therefore the inertial forces can neglected and the

equation of motion of the medium reduces to the quasi- static balance of forces:

_𝜕𝑥 ^𝜕 _𝑗 𝜎 _𝑖𝑗 + 𝐹 _𝑖 = 0

In the case of tissue F can represent forces exerted on the cytogel by extracellular matrix consisting of a net of actin fibers.

E.g. we can assume that F is a restoring force proportional

to the displacement vector-field u(x).

F= -ku

Diffusion in the whole ℝ ^𝟑

Mechanical part. We assume that the traction tensor is isotropic,

𝜏 _𝑖𝑗 = 𝜏𝛿 _𝑖𝑗 . Expressing the strain tensor in terms of the displacement vector field 𝑢 (𝑥),

𝜖 _𝑖𝑗 ≝ ¹ ₂ (𝑢 _𝑖,𝑗 + 𝑢 _{𝑗 ,𝑖} ),

we can write balance of forces as

∇ 𝜇 + 𝜆 𝜃 + 𝜈 ₁ 𝜃 + ¹

2 𝜈 ₂ 𝑑𝑖𝑣𝑢 + 𝜏 + 𝜇Δ𝑢 + ¹

2 𝜈 ₂ Δ𝑢 = k𝑢

To solve this equation we can introduce the potential for the

displacement vector-field 𝑢 i.e. 𝑢 𝑥 = 𝑔𝑟𝑎𝑑𝜓. Then, denoting

𝜈 ≝ 𝜈 ₁ + 𝜈 ₂

and noticing that 𝜃 = Δ𝜓 we obtain :

𝛻 2𝜇 + 𝜆 𝜃 + 𝜈𝜃 + 𝜏 − 𝑘𝜓 = 0

Assuming that the stresses are vanishing at infinity we have

2𝜇 + 𝜆 𝜃 + 𝜈𝜃 + 𝜏 = 𝑘𝜓

By the definition of 𝜃 we obtain the following linear evolutionary equation for the displacement potential

𝜈 _𝜕𝑡 ^𝜕 ∆𝜓 + 2𝜇 + 𝜆 ∆𝜓 − 𝑘𝜓 + 𝜏(𝑐) = 0

In general the viscous forces are much smaller than the elastic ones, so they can treated as a perturbation, thus writing

𝜖 _𝑖𝑗 = 𝜖 _𝑖𝑗 ⁽⁰⁾ + 𝜈𝜖 _𝑖𝑗 ⁽¹⁾ + ⋯ ,

and consequently

𝜃 = 𝜃 ⁽⁰⁾ + 𝜈𝜃 ⁽¹⁾ + ⋯ and 𝜓 = 𝜓 ⁽⁰⁾ + 𝜈𝜓 ⁽¹⁾ + ⋯

For the first approximation we obtain (case k=0)

2𝜇 + 𝜆 𝜃 ⁽⁰⁾ + 𝜏 = 𝑘𝜓 ⁽⁰⁾ ,

which is equivalent by definition of θ to the following Helmholtz equation

Δ𝜓 ⁽⁰⁾ − _{2𝜇 +𝜆} ^𝑘 𝜓 ⁽⁰⁾ = − ¹

2𝜇 +𝜆 𝜏

whereas for the first correction terms we obtain

2𝜇 + 𝜆 𝜃 ⁽¹⁾ − 𝑘𝜓 ⁽¹⁾ + 𝜈𝜃 ⁽⁰⁾ = 0, so 𝜃 ⁽¹⁾ − _{2𝜇 +𝜆} ^𝑘 𝜓 ⁽¹⁾ = − ¹

(2𝜇 +𝜆) ² 𝜏

Similarly as before we obtain the Helmholtz equation for the first correction for

𝛥𝜓 ⁽¹⁾ − _{2𝜇 +𝜆} ^𝑘 𝜓 ⁽¹⁾ = ¹

(2𝜇 +𝜆) ² 𝜏

The asymptotic analysis of equations (10) and (12) for large and

small k is possible.

Here we consider here mainly the case of k=0. In this case we

have 𝜃 ⁽⁰⁾ = − ^𝜏

2𝜇 +𝜆 and 𝜃 ⁽¹⁾ = ¹

(2𝜇 +𝜆) ² 𝜏

2.2 Reaction diffusion equation. The calcium diffusion equation with incorporated mechanical effects is

𝑐 _𝑡 = 𝐷𝛥𝑐 + 𝑓(𝑐, 𝜃)

The experimental determination of the source function f(c,θ) seems to be very difficult, especially that the proposed

mathematical model should be treated as an approximation of much more complex reality. The source term f(c,θ) should satisfy some physical restrictions e.g. it should be nonnegative when concentration c approaches zero. In [1] f is

proposed as a sum:

f(c,θ) = f(c) + γθ,

which can be thought of as a first term in the power series of f with respect to the dilation θ. This form for large values θ can

however lead to unphysical behavior- the source term can become negative for c=0, thus leading to unphysical negative values of calcium concentrations. Still it can be useful in explaining some interesting features of the mechno-chemical coupling. The

experiments in which calcium waves are generated by squeezing locally the tissue or part of a cell see[ ], can be explained in the frame of discussed here mechano-chemical model if the coupling constant γ is negative. Indeed squeezing induces negative θ,

therefore if initially the system was in the lower stable state C

1

(ground state) then when θ decreases during the gradual

squeezing, the ground state C

1

(θ) increases till the moment where the minimum of the function f(c,θ)+γθ with respect to c becomes equal to zero. At this moment the stable and unstable states

merge and C

₁

becomes unstable for positive perturbations of c.

When θ is still increasing the state C

1

(as well as C

2

) cease to exist and the system is jumping to the upper stable state. In this way the local concentration becomes high enough to start the

propagation of travelling wave supported in its further evolution by the autocatalytic mechanism, encoded in the model by

bistability of f(c).

In our appr. 𝜃 = 𝜃 ⁰ + 𝜃 ¹ . Assuming that, 𝜃 ⁰ 𝑐 = − _{2𝜇 +𝜆} ^𝜏(𝑐) is already encoded In the form of f ; that is assuming that

𝑓 𝑐, 𝜃 = 𝑓(𝑐) + 𝛾𝜃 ⁰ 𝑐 + 𝛾𝜃 ⁽¹⁾ = 𝑔 𝑐 + 𝛾𝜃 ⁽¹⁾ we obtain

𝑐 _𝑡 = 𝐷𝛥𝑐 + 𝑔 𝑐 + 𝛾𝜃 ⁽¹⁾ or

𝑐 _𝑡 = 𝐷𝛥𝑐 + 𝑔 𝑐 + 𝛾 _{(2𝜇 +𝜆)} ^𝜈 ₂ 𝜏

Noticing that 𝜏 = 𝜏, _𝑐 𝑐 _𝑡 we finally arrive at a single reaction diffusion equation

𝛽(𝑐)𝑐 _𝑡 = 𝐷𝛥𝑐 + 𝑔 𝑐 ,

where 𝛽 = (1 − 𝛾 _{(2𝜇 +𝜆)} ^𝜈 ₂ 𝜏, _𝑐 ) .

The existence of travelling waves solutions follows immediately from the appropriate theorem for a single reaction diffusion equation, provided that g(c) is of a bi-stable type. In the case when β is a non-vanishing constant or does not change much, the viscosity influences only the speed of the wave leaving the wave profile unchanged.

For very large k, when other terms except of 𝜓 and 𝜏 can be neglected we have 𝜓 = −𝑘 ⁻¹ 𝜏 and then the reaction diffusion equation for calcium concentration becomes

𝑐 _𝑡 = 𝐷 − 𝛾𝑘 ⁻¹ 𝜏 _,𝑐 𝛥𝑐 + 𝑓 𝑐 − 𝛾𝑘 ⁻¹ 𝜏 _,𝑐𝑐 (∇𝑐) ² .

3.Diffusion in a thin layer of a visco-elastic material

Consider a thin layer of visco-elastic material; 𝑥 ¹ , 𝑥 ² , 𝑥 ³ ∈ ℜ ² × [−𝑑, 𝑑] with free, unloaded upper and lower surfaces. Under the influence of internal stresses such layer can in principle undergo buckling. This possibility, however, will not be analyzed here. In this Section we assume that the traction tensor 𝜏 can be somewhat anisotropic, i.e. it can be of the following form

𝜏 =

𝜏 0 0

0 𝜏 0

0 0 𝜏 ₃₃

Together with 𝑥 ¹ , 𝑥 ² , 𝑥 ³ we use here x, y, z and 𝑥 = (𝑥 ¹ , 𝑥 ² ) for the convenience.

Since the layer is thin we can expand the displacement vector-field in powers of z (=x

³

) up to second order terms (the dependence on t is suppressed for the convenience)

𝑢 _𝑖 = 𝑎 _𝑖 ⁰ 𝑥 + 𝑎 _𝑖 ⁽¹⁾ (𝑥 )𝑧 + 𝑎 _𝑖 ⁽²⁾ (𝑥 )𝑧 ²

The plane 𝑥 ¹ , 𝑥 ² , 0 is assumed to be a symmetry plane, so we have

𝑢 _𝛼 𝑥 , 𝑧 = 𝑢 _𝛼 𝑥 , 𝑧 , α = 1,2

𝑢 ₃ 𝑥 , −𝑧 = −𝑢 ₃ 𝑥 , 𝑧

This implies: 𝑢 ₃ = 𝑎 ₃ ¹ 𝑧 and 𝑢 _𝛼 ¹ ≡ 0, so finally

𝑢 _𝛼 = 𝑎 _𝛼 ⁰ 𝑥 + 𝑎 ² (𝑥 )𝑧 ²

𝑢 ₃ = 𝑎 ₃ ¹ 𝑥 𝑧

Consequently the strain tensor is given by : 𝜖 _𝛼𝛽 = ¹

2 {𝑎 _𝛼,𝛽 ⁰ + 𝑎 _𝛼,𝛽 ² 𝑧 ² + 𝑎 _𝛽,𝛼 ⁰ + 𝑎 _𝛽,𝛼 ² 𝑧 ² } 𝜖 _𝛼3 = ¹

2 2𝑎 _𝛼 ² + 𝑎 _3,𝛼 ¹ 𝑧 𝜖 ₃₃ = 𝑎 ₃ ¹

Boundary conditions. Since the top and bottom surfaces are free unloaded surfaces the following relations must be satisfied for 𝑧 = ±𝑑:

1. 𝜎 _𝛼3 = 0 : , 2𝜇𝜖 _𝛼3 + 𝜈 ₂ 𝜖 _𝛼3 = 0 2. 𝜎 ₃₃ = 0 : 𝜆𝜃 + 2𝜇𝜖 ₃₃ + 𝜈 ₁ 𝜃 + 𝜈 ₂ 𝜖 ₃₃ + 𝜏 ₃₃ = 0

Balance of forces , 𝜙(𝑡, 𝑥, 𝑦) potential:

(𝜆 + 𝜇)𝜃 − 𝜇𝜖 ₃₃ + 𝜈𝜃 − 𝜈 ₂ 𝜖 ₃₃ + 𝜏 − 𝑘𝜙 = 0

Boundary conditions: If 𝑧 = ±𝑑 (free unloaded surfaces) we require that for i= 1,2,3, 𝜎𝑖3 =0; thus for i=α=1,2 we arrive at

𝜖

_𝛼3

|

_z=±𝛿

= 0 ⇒ 2𝑎

₂ ²

+ 𝑎

_3,𝛼 ¹

= 0 ⇒ 𝜖

_𝛼3

= 0 (23a)

Whereas for i=3 we have

𝜆𝜃 + 2𝜇𝑎

₃⁽¹⁾

+ 𝜏

₃₃

= 0 (23b)

Let us note that If we do the averaging of Eqs (22) over the layer thickness 2d, we obtain similar results 𝜖𝛼𝛽 =¹₂(𝑎_𝛼,𝛽+ 𝑎_{𝛽 ,𝛼}) +₂¹(𝑎_𝛼,𝛽 ² + 𝑎_{𝛽 ,𝛼} ² )^𝑑₃² (24a)

𝜖𝛼3 = 0 , 𝜖 = 𝑎₃⁽¹⁾ (24b)

Now we can proceed in a similar way as previously, assuming that 𝜏 depends only on c, whereas c can be a function of t,x,y, but not z, we introduce the two-dimensional potential 𝜓, 𝑢𝛼 =_𝜕𝑥^𝜕_𝛼𝜓 ,

ale 𝜃 = 𝜖11+ 𝜖₂₂ + 𝜖₃₃ = 𝑎₁₁+ 𝑎₂₂ + 𝑎₃¹ So in zero order approximation:

𝜆 𝑎₁₁ + 𝑎₂₂ + 2𝜇 + 𝜆 𝑎₃ ¹ + 𝜏₃₃ = 0 Propagation in thin layers – plane stress state assumption.

𝜎

_𝑖𝑗

= 𝜎

_𝛼𝛽

, 𝜎

_𝛼3

, 𝜎

₃₃

𝛼, 𝛽 = 1,2 𝜃 = 𝜃 + 𝜖

₃₃

gdzie 𝜃 = 𝜖

₁₁

+ 𝜖

₂₂

,

Boundary conditions. Since the top and bottom surfaces are free unloaded surfaces the

following relations must be satisfied for 𝑧 = ±𝑑:

1. 𝜎

_𝛼3

= 0 , which implies 2𝜇𝜖

𝛼3

= 0 ( ) 2. 𝜎

₃₃

= 0 which implies 𝜆𝜃 + 2𝜇𝜖

₃₃

+ 𝜏

₃₃

= 0 at 𝑧 = ±𝑑 ( ) Let us note that:

1. If the calculations are made up to main, i.e. zero order terms in d, then the coefficients of the strain tensor are depending only x and y. Indeed we have

𝜖_𝛼𝛽 =¹

2(𝑎_𝛼,𝛽 + 𝑎_𝛽,𝛼) 𝜖_𝛼3= 0 , 𝜖₃₃ = 𝑎₃⁽¹⁾

2. If second order terms in d are preserved, then as it can be easily verified, the averaging over the layer thickness in the mechanical equilibrium equations, denoted here by < >, commutes with differentiation

<_𝜕𝑥^𝜕_𝑗

𝜎

_𝑖𝑗

>=

_𝜕𝑥^𝜕_𝑗

< 𝜎

_𝑖𝑗

>

Therefore, mechanical equilibrium equations

𝜎

𝑖𝑗 ,𝑗

= 𝑘 𝑢, where 𝑘 = 𝑘 0 0 0 𝑘 0

0 0 0 ,

are reduced to

𝜎

_{𝛼𝛽 ,𝛽}

= 𝑘𝑢

_𝛼

, with

𝛼, 𝛽 = 1,2

.

Now it is convenient to introduce a 2-dimensional dilation,

𝜃 =

𝜖₁₁+ 𝜖₂₂ , related to the two-dimensional strain tensor

𝜖

_𝛼𝛽

,

𝛼, 𝛽 = 1,2. The boundary condition

𝜎

₃₃

= 0

gives us

𝜆𝜃 + 2𝜇 + 𝜆 𝜖

₃₃

+ 𝜏

₃₃

= 0 ⇒ 𝜖

₃₃

= −

_{2𝜇 +𝜆}¹

{𝜏

₃₃

+ 𝜆𝜃 } Proper b-dry condition is:

𝜆𝜃 + 2𝜇 + 𝜆 𝜖

₃₃

+ 𝜈

₁

𝜃 + 𝜈

₂

𝜖

₃₃

+ 𝜏

₃₃

= 0 A. Mechanical equilibrium equations

𝜎

_{𝑖𝑗 ,𝑗}

= 0 ⇒ 𝜎

_{𝛼𝑗 ,𝑗}

= 0 ⇒ 𝜎

_{𝛼𝛽 ,𝛽}

= 0

Since all quantities in the above equations are independent of z, i.e. they depend on 𝑥

¹

, 𝑥

²

and 𝑡 then

𝜆𝜃𝛿

_𝛼𝛽

+ 2𝜇𝜖

_𝛼𝛽

+ 𝜏

_{𝛼𝛽 ,𝛽}

= 0 Full equation of mechanical equilibrium (with viscous terms) is 𝜆𝜃𝛿

_𝛼𝛽

+ 2𝜇𝜖

_𝛼𝛽

+ 𝜈

₁

𝜃 𝛿

_𝛼𝛽

+ 𝜈

₂

𝜖

_𝛼𝛽

+ 𝜏

_{𝛼𝛽 ,𝛽}

= 0, 𝜆𝜃 𝛿

_𝛼𝛽

+ 𝜆𝜖

₃₃

𝛿

_𝛼𝛽

+ 2𝜇𝜖

_𝛼𝛽

+ 𝜈

₁

𝜃 𝛿

_𝛼𝛽

+ 𝜈

₁

𝜖

₃₃

𝛿

_𝛼𝛽

+ 𝜈

₂

𝜖

_𝛼𝛽

+ 𝜏

_𝛼𝛽

,𝛽

= 0 which can be transformed to (nie jest dobrze bo eps33 jest policz niedokl)

(𝜆 + 𝜇)𝜃 + 𝜆𝜖

₃₃

+ 𝜈𝜃 + 𝜈

₁

𝜖

₃₃

+ 𝜏 − 𝑘𝜓 = 0

2𝜇𝜆

2𝜇+𝜆

𝜃 𝛿

_𝛼𝛽

+ 2𝜇𝜖

_𝛼𝛽

+ 𝜈

₁

𝜃 𝛿

_𝛼𝛽

+ 𝜈

₂

𝜖

_𝛼𝛽

−

_{2𝜇 +𝜆}^𝜈1

𝜏

₃₃

+ (𝜏 −

_{2𝜇 +𝜆}^𝜆

𝜏

₃₃

)

,𝛽

= 𝑘𝑢

_𝛼 Using the potential for u, and denoting as before 𝜈 = 𝜈1+ 𝜈₂, we obtain after integration we again arrive at a Helmholtz equation for the potential

𝜓(𝑡, 𝑥, 𝑦)

(

_2𝜇+𝜆^2𝜇𝜆

+ 𝜇)Δ𝜓 + 𝜈Δ𝜓 − 𝑘𝜓 + 𝜏 −

_{2𝜇 +𝜆}^𝜆

𝜏

₃₃

−

_2𝜇+𝜆^𝜈1

𝜏

₃₃

= 0 From now on we assume k=0, and expand the potential as before

𝜇 2𝜇 +𝜆

2𝜇 +𝜆

Δ𝜓 + 𝜈Δ𝜓 − 𝑘𝜓 + 𝜏 −

_{2𝜇 +𝜆}^𝜆

𝜏

₃₃

−

_2𝜇+𝜆^𝜈1

𝜏

₃₃

= 0

2𝜇𝜆

2𝜇+𝜆

𝜃 𝛿

_𝛼𝛽

+ 2𝜇𝜖

_𝛼𝛽

+ (𝜏 −

_{2𝜇 +𝜆}^𝜆

𝜏

₃₃

)

,𝛽

= 0

2𝜇𝜆

2𝜇 +𝜆

𝜃 𝛿

_𝛼𝛽

+ 2𝜇𝜖

_𝛼𝛽

+ 𝜏 −

_2𝜇+𝜆^𝜆

𝜏

₃₃

𝛿

_𝛼𝛽

,𝛽

= 𝑘𝑢

_𝛼

where 𝜃 = 𝜃 + 𝜖

₃₃

.Introducing (2-dimensional) potential: 𝑢

_𝛼

= 𝜙

_,𝛽

we arrive at

∇

_α

4μ

_{2𝜇 +𝜆}^{λ +μ}

Δ𝜙 + 𝜏 −

_{2𝜇 +𝜆}^𝜆

𝜏

₃₃

− 𝑘𝜙 = 0,

which after integration gives

4μ

_{2𝜇 +𝜆}^{λ +μ}

Δ𝜙 + 𝜏 −

_{2𝜇 +𝜆}^𝜆

𝜏

₃₃

− 𝑘𝜙 = 0

4 Diffusion in a long thin fiber.

In this case we assume that traction tensor is symmetric with respect to x (= x¹ ) axis

𝜏 =

𝜏₁₁ 0 0

0 𝜏 0

0 0 𝜏

By the axial symmetry of the problem we have

𝜎_𝑖𝑗 = 𝜎11, 𝜎_1𝛼, 𝜎_𝛼𝛽 𝛼, 𝛽 = 2,3 Z symetrii 𝜖22 = 𝜖₃₃, a więc 𝜃 = 𝜃 + 2𝜖22 𝜃 = 𝜖11 + 2𝜖₂₂

Warunki brzegowe 𝜎

_𝑖2

= 0, 𝜎

_𝑖3

= 0

i=2

𝜆𝜃 + 2𝜇𝜖

₂₂

+ 𝜏 = 0 𝜃 = 𝜃 + 2𝜖

₂₂

stąd

𝜆𝜃 + 2 𝜆 + 𝜇 𝜖

₂₂

+ 𝜏

₂₂

= 0 → 𝜆𝜖

₁₁

+ 2 𝜆 + 𝜇 𝜖

₂₂

+ 𝜏

₂₂

= 0

Równania równowagi:

𝜆𝜃_,1+ 2𝜇𝜖_11,1+ 𝜏_11,1= 0 𝜆𝜃 + 2𝜇𝜖₁₁+ 𝜏₁₁ = 0 stąd

2𝜇 + 𝜆 𝜖

₁₁

+ 2𝜆𝜖

₂₂

+ 𝜏

₁₁

= 0 𝜆𝜖

₁₁

+ 2 𝜆 + 𝜇 𝜖

₂₂

+ 𝜏

₂₂

= 0

𝜖₂₂ =𝜆𝜏₁₁− 𝜆 + 2𝜇 τ

2𝜇 2𝜇 + 3𝜆 = − 1 2𝜇 + 3𝜆𝜏

𝜖₁₁ =𝜆𝜏 − 𝜆 + 𝜇 𝜏₁₁

𝜇 2𝜇 + 3𝜆 = − 1

2𝜇 + 3𝜆𝜏

In the case of isotropic traction tensor

𝜏

₁₁

= 𝜏,

we have

𝜖

₁₁

= 𝜖

₂₂

= 𝜖

₃₃and 𝜃 = 3𝜖11 = −3 ¹

2𝜇+3𝜆𝜏

Uwaga: można oczywiście postulowad równanie 𝑓 = 𝑓(𝑐. 𝜃, 𝜃) wtedy 𝜃 - wylicza się przez c.

---

………

𝛻 𝜇 + 𝜆 𝜃 + 𝜈1𝜃 +1

2𝜈₂𝑑𝑖𝑣𝑈 + 𝜏 + 𝜇𝛥𝑢 +1

2𝜈₂𝛥𝑢 = 0

Thus to solve this equation we can introduce the potential for 𝑢 i.e. 𝑢 𝑥 = 𝑔𝑟𝑎𝑑𝜓. Then noticing that 𝜃 = Δ𝜓 we obtain:

𝛻 2𝜇 + 𝜆 𝜃 + (𝜈₁+ 𝜈₂)𝜃 + 𝜏 = 0 Thus we have:

1. 2𝜇 + 𝜆 𝜃

⁽⁰⁾

+ 𝜏 = 0 Δ𝜓

⁽⁰⁾

= − 𝜏

2𝜇 + 𝜆 2. 2𝜇 + 𝜆 𝜃

⁽¹⁾

+ 𝜈𝜃 = 0 So Δ𝜓

⁽¹⁾

=

_{(2𝜇 +𝜆)}^𝜈 ₂

𝜏 and 𝜃

⁽¹⁾

= −

_{2𝜇 +𝜆}^𝜈

𝜃

⁽⁰⁾

………

By taking the trace of the last equation we obtain

3𝜆 + 2𝜇 𝜃 ¹ + 3𝜈₁+ 𝜈₂ 𝜃 ⁰ = 0 and consequently 𝜃⁽¹⁾= −^3𝜈_3𝜆+2𝜇^1+𝜈2𝜃 ⁽⁰⁾.

By taking the traceless part of equation 2. we arrive at

2𝜇 𝜖⁽¹⁾−¹

3𝜃⁽¹⁾𝛿_𝑖𝑗 + 𝜈₂ 𝜖 _𝑖𝑗 ⁰ 𝛿_𝑖𝑗 = 0,

which finally allows us to find the correction to the strain tensor due to the viscosity 𝜖_𝑖𝑗⁽¹⁾= 𝜈₂

2𝜇 1

3𝜃 ⁰ 𝛿_𝑖𝑗 − 𝜖 _𝑖𝑗 ⁰ −1 3

3𝜈₁+ 𝜈₂ 3𝜆 + 2𝜇𝜃 ⁰ 𝛿_𝑖𝑗

In the case of isotropic traction tensor, (𝜏𝑖𝑗 = 𝜏𝛿_𝑖𝑗) , Eq.(1) after introducing the displacement vector- field u(x) gives us

𝜇 + 𝜆 𝛻𝜃⁽⁰⁾+ 𝜇𝛥𝒖 + 𝛻𝜏 = 0

for the zero-order approximation. In general the displacement u(x) can be decomposed into the sum of the gradient of a potential and the curl of a certain vector-field. As all other terms in the equation are gradient-like and the material in the infinity is assumed to be undisturbed, then it is sufficient to introduce the displacement potential 𝒖 = ∇ψ. In this way we come to

𝛁 𝟐𝝁 + 𝝀 𝚫𝝍 + 𝝉 = 𝟎

So, up to a constant we have

∆𝝍 = − 𝝉 𝟐𝝁 + 𝝀

Noticing that Δ𝜓 = 𝜃 we have 𝜃⁽⁰⁾= − ^𝜏

2𝜇 +𝜆.

Developing the potential Ψ similarly as the strain tensor

𝜓 = 𝜓⁽⁰⁾+ 𝜓⁽¹⁾+ ⋯ we have 𝛥𝜓⁽¹⁾= −𝜃⁽¹⁾

………

we have 𝛥𝜓⁽¹⁾= 𝜃⁽¹⁾

where the zero-part of the strain tensor corresponds to the vanishing viscosities. Comparing the terms of the same order we obtain

1. 𝜎

⁽⁰⁾

= 𝜆𝜃

⁽⁰⁾

𝛿

_𝑖𝑗

+ 2𝜇𝜖

_𝑖𝑗⁽⁰⁾

+ 𝜏

_𝑖𝑗

2. 𝜎

⁽¹⁾

= 𝜎𝜆𝜃

⁽¹⁾

𝛿

_𝑖𝑗

+ 2𝜇𝜖

_𝑖𝑗⁽¹⁾

+ 𝜈

₁

𝜃

⁽⁰⁾

𝛿

_𝑖𝑗

+ 𝜈

₂

𝜖

_𝑖𝑗⁽⁰⁾

………

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